A random sample of 150 investment bankers (population 1) had thirty percent who expressed dissatisfaction with their choice of career. A random sample of 200 financial analysts (population 2) had forty-two percent who expressed dissatisfaction. In testing the hypothesis that a higher proportion of financial analysts than investment bankers are dissatisfied, the standard deviation of the difference in the two population proportions is approximated by ______________ and the standardized test statistic is ______________.

Question 13 options:

.0027; -2.303


.0027; -44.44


.0521; -2.303


.0521; -44.44

To find the standard deviation of the difference in the two population proportions and the standardized test statistic, you can follow these steps:

Step 1: Calculate the standard deviation of the difference in the two population proportions.
The formula for the standard deviation of the difference in proportions is:

σ = sqrt((p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2))
where
σ = standard deviation of the difference in proportions
p1 = proportion of population 1
p2 = proportion of population 2
n1 = size of sample from population 1
n2 = size of sample from population 2

Given information:
p1 = 0.3 (30% expressed dissatisfaction among investment bankers)
p2 = 0.42 (42% expressed dissatisfaction among financial analysts)
n1 = 150 (sample size for investment bankers)
n2 = 200 (sample size for financial analysts)

Substituting the values:
σ = sqrt((0.3 * (1 - 0.3) / 150) + (0.42 * (1 - 0.42) / 200))
= sqrt((0.21 / 150) + (0.2457 / 200))
= sqrt(0.0014 + 0.00123)
= sqrt(0.00263)
≈ 0.0513 (approximately)

Therefore, the standard deviation of the difference in the two population proportions is approximately 0.0513.

Step 2: Calculate the standardized test statistic.
The standardized test statistic is given by the formula:

z = (p1 - p2) / σ
where
z = standardized test statistic
p1 = proportion of population 1
p2 = proportion of population 2
σ = standard deviation of the difference in proportions

Given information:
p1 = 0.3 (30% expressed dissatisfaction among investment bankers)
p2 = 0.42 (42% expressed dissatisfaction among financial analysts)
σ ≈ 0.0513 (from step 1)

Substituting the values:
z = (0.3 - 0.42) / 0.0513
= -0.12 / 0.0513
≈ -2.33 (approximately)

Therefore, the standardized test statistic is approximately -2.33.

Hence, the correct option is:
.0521; -2.303